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Why curve is 1-dimensional object?

In "OpenGIS® Implementation Standard for Geographic information - Simple feature access - Part 1: Common architecture" is stated:

A Curve is a 1-dimensional geometric object that is the homeomorphic image of a real, closed, interval in the coordinate space.

Looking at definition of homeomorphism:

A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions

and taking as an example a LinearRing that is a LineString (which is a Curve with linear interpolations between points) which has a common point of the start (s) of the starting line segment and the end (e) of the ending line segment I cannot understand or prove to myself that a LinerRing is a homeomorphic image of an interval.

Any help is highly appreciated.

UPDATE:

I have read the definitions more carefully (Wikipedia) and they have clarified the situation.

  1. By definition a curve is a continuous (not homeomorphic!) mapping from interval to a topological space
  2. If a mapping is homeomorphic then the curve is called simple
  3. By convention if an interval start and end are mapped to the same curve point, then the curve is called closed (or a loop). A closed curve is a continuous mapping of a circle.

If it is defined so then I may conclude: the curve is only 1-dimensional when there is a homeomorphism from interval to the topological space, a ring cannot be mapped in this way and therefore is not 1-dimensional. Moreover, not every curve is 1-dimensional.

The OpenGIS document does not define the closed curve (or a ring) explicitly and therefore the text where it is written is confusing. My confusion was mainly connected with the following logical consequence: 1) A curve is a homeomorphism from interval to a coordinate space therefore 2) the curve is 1-dimensional. 3) A ring is a curve with starting and ending point of an interval mapped to the same point (closed curve) and 4) since a ring is a simple (there is not intersections) and closed curve then it is 1-dimensional. In fact, it is nowhere stated in the document that a closed curve is 1-dimensional. I understood that when I have explicitly found definition of the closed curve.

Solution

  • Typically, a curve is one-dimensional because you only need a single number to describe a point's position on the curve: distance from an end point or chosen origin.

    Describing the space the curve occupies in a larger world is another matter completely :) but you could place a point on a straight number line, a point on the curve, and for every unit of movement of one point, move the other point a corresponding distance.